From a point x = 110 feet in front of a public library, the angles of elevation to the base of the flagpole and the top of the flagpole are θ = 27.5° and 39° 45', respectively. The flagpole is mounted on the front of the library's roof. Find the height of the flagpole.

STEP 1: Let F be the height of the flagpole and L be the height of the public library. Convert the angle 39° 45' into decimal form.
Use this value and an appropriate trigonometric function to find F + L, the total height of the library and flagpole.
F + L=

STEP 2: Find the height of the library, L.

STEP 3: Subtract your result from Step 2 from your latter result in Step 1 to determine the height of the flagpole, F.

39° 45' = ( 39 + 45 / 60 )° = 39.75°

tan 39.75° = ( F + L ) / 110

F + L = 110 ∙ tan 39.75°

F + L = 110 ∙0.831691221 = 91.48603431 ft

height of the library:

tan 27.5° = L / 110

L = 110 ∙ tan 27.5° = 110 ∙ 0.520567051 = 57.26237561 ft

height of the flag:

F = F + L - L = 91.48603431 - 57.26237561 = 34.2236587 ft

Well, follow the directions

tan (39 + 45/60) = (F+L)/110 find F+L
tan (27.5) = L/110 find L
so subtract

STEP 1: Let F be the height of the flagpole and L be the height of the public library. Convert the angle 39° 45' into decimal form.

Use this value and an appropriate trigonometric function to find F + L, the total height of the library and flagpole.
F + L = tan(39° 45') * 110 feet

STEP 2: Find the height of the library, L.
Since the angle of elevation to the base of the flagpole is given as 27.5°, we can use the tangent function to find L.
L = tan(27.5°) * 110 feet

STEP 3: Subtract your result from Step 2 from your latter result in Step 1 to determine the height of the flagpole, F.
F = (tan(39° 45') * 110 feet) - (tan(27.5°) * 110 feet)

Perform these calculations to find the final answer.

STEP 1: To find F + L (the total height of the library and flagpole), we can use the tangent function since we have the angle of elevation and the distance from the point x to the flagpole's base.

We know that tan(θ) = opposite/adjacent, where θ is the angle of elevation and opposite is the height of the flagpole.

Using θ = 39° 45', we need to convert it to decimal form. To do that, we convert the minutes to degrees by dividing 45 by 60: 45/60 = 0.75 degrees. Adding this to the original angle: 39 + 0.75 = 39.75 degrees.

Now we can use the tangent function to find the height of the flagpole, F. We have the opposite side (F) and the adjacent side (110 feet). So we have the equation:

tan(39.75) = F/110

Solving for F, we multiply both sides of the equation by 110:

F = 110 * tan(39.75)

Now we have the value of F.

STEP 2: To find the height of the library, L, we can use the same approach. From the given information, we have the angle of elevation θ = 27.5° and the distance from the point x to the flagpole's base (110 feet). Again, we use the tangent function:

tan(27.5) = L/110

Solving for L, we multiply both sides of the equation by 110:

L = 110 * tan(27.5)

Now we have the value of L.

STEP 3: To find the height of the flagpole, we subtract the height of the library (L) from the total height of the library and flagpole (F + L):

Height of the flagpole, F = (F + L) - L

Substituting the values we found in steps 1 and 2:

F = (110 * tan(39.75)) - (110 * tan(27.5))

Now we can calculate the height of the flagpole, F.